Apparatus for evaluating a polynomial function using an array of optical modules

ABSTRACT

Methods and apparatus for providing an optical analog quantity proportional to ##EQU1## Input light (a 4 ) of intensity proportional to a n  is directed (21,31) to the input end of an nth optical module comprising a modulator (22) whose output light intensity is responsive to an electrical potential difference across it and a beam combiner (23). While the light is passing through the modulator (22), a potential difference (24) is applied across it such that the intensity of the output light from the modulator (22) is proportional to a n  x, and this light (a 4  x) is directed, through the beam combiner (23), to the output end of the optical module. Input light (a 3 ) of intensity proportional to a n-1  is directed (25), via the beam combiner (23), to the output end of the optical module, to combine with the light from the modulator (22) so that the intensity of the output light (32) from the nth optical module is proportional to a n  x+a n-1  ; this light is directed to the input end of an (n-1)th optical module essentially similar to the nth; and so on in the same manner; and finally to the first module, so that its output light has an intensity proportional to p(x).

FIELD

This invention relates to systolic pipeline processing for evaluation ofpolynomial functions with optical methods and apparatus. It isespecially useful for analog computation at very high speeds, and has awide variety of applications.

BACKGROUND

The present invention is related to the subject matter of the copendingU.S. patent application Ser. No. 450,153, filed Dec. 15, 1982, of HenryJohn Caulfield, for Systolic Array Processing, which discloses subjectmatter generic to some of the matter in the present application; andSer. No. 459,168, filed Jan. 19, 1983 (the same date on which thepresent application was filed), of H. J. Caulfield, for PolynomialEvaluation, which discloses most of the subject matter in the presentapplication. Said applications are assigned to the assignee of thepresent invention. Also said earlier application is hereby incorporatedhereinto by reference and made a part hereof the same as if fully setforth herein, for purposes of indicating the background of the presentinvention and illustrating the state of the art.

Except where otherwise indicated herein, the electrooptic componentsemployed in typical embodiments of the present invention are now wellknown. Convenient ways of making them are described in the abovementioned patent applications and in the references cited therein andherein.

The disclosure in the earlier copending application includes the paperH. J. Caulfield, et al.sup.(2) wherein it is shown how certainalgorithms for matrix-vector multiplication can be implemented usingacoustooptic cells for multiplication and input data transfer and usingCCD detector arrays for accumulation and output of the results. No 2-Dmatrix mask is required; matrix changes are implemented electronically.A system for multiplying a 50-component nonnegative-real matrix isdescribed. Modifications for bipolar-real and complex-valued processingare possible, as are extensions to matrix-matrix multiplication andmultiplication of a vector by multiple matrices.

During the last several years, Kung and Leierson at Carnegie-MellonUniversity.sup.(1,5) have developed a new type of computationalarchitecture which they call "systolic array processing". Although thereare numerous architectures for systolic array processing, a generalfeature is a flow of data through similar or identical arithmetic orlogic units where fixed operations, such as multiplications andadditions, are performed. The data tend to flow in a pulsating manner,hence the name "systolic". Systolic array processors appear to offercertain design and speed advantages for VLSI implementation overprevious calculational algorithms for such operations as matrix-vectormultiplication, matrix-matrix multiplication, pattern recognition incontext, and digital filtering. The earlier copending application dealswith improving systolic array processors by using optical input andoutput as well as new architectures for optical signal processing,particularly for multiplications involving at least one matrix; and itpoints out that many other operations ca be performed in an analogousmanner.

DISCLOSURE

The present invention comprises methods and apparatus for providingoptical analog quantities for evaluating polynomial functions.

A typical method according the present invention for providing anoptical analog quantity that is approximately proportional to the valueof a polynomial function expressible in the form ##STR1## comprisesdirecting input light of intensity approximately proportional to thecoefficient a_(n) to the input end of an nth optical module comprisingat the input end a modulator whose output light intensity isapproximately proportional to a known function of an electricalpotential difference across it and at the output end a beam combiner,applying across the modulator, while the light is passing through it, apotential difference approximately proportional to a function of x suchthat the intensity of the output light from the modulator isapproximately proportional to a_(n) x, directing the output light fromthe modulator, through the beam combiner, to the output end of theoptical module, directing input light of intensity approximatelyproportional to the coefficient a_(n-1), via the beam combiner, to theoutput end of the optical module, to combine with the output light fromthe modulator so that the intensity of the output light from the nthoptical module is approximately proportional to a_(n) x+a_(n-1) ;directing the output light from the nth optical module to the input endof an (n-1)th optical module essentially similar to the nth; and so onin the same manner; and finally to the first module, so that its outputlight has an intensity approximately proportional to p(x).

Typical apparatus according to the present invention for performing themethod comprises an electrooptic planar waveguide; a plurality, n, ofintegrated optical modules in the waveguide, each module comprising atits input end a modulator for receiving light travelling in apredetermined input direction and transmitting a controlled proportionthereof further in a predetermined output direction to the output end,and a beam combiner for receiving the light from the modulator andtransmitting a predetermined proportion thereof further in thepredetermined output direction and on through the output end of themodule;

the modulator comprising electrooptic diffractive means comprising apair of electrodes for forming a Bragg grating therebetween, positionedwith a direction of Bragg incidence approximately in the predeterminedinput direction, the first electrode comprising a first set ofsubstantially straight and parallel, thin, elongate, electricallyconductive members connected together at one end, and the secondelectrode comprising a second set of substantially straight andparallel, thin, elongate, electrically conductive members, interleavedwith the first set, insulated therefrom, and connected together at theopposite end, so that when a first electrical potential is applied tothe first electrode and a second electrical potential is applied to thesecond electrode the controlled portion of input light transmittedthrough the modulator is provided by a Bragg diffraction thereof in thepredetermined output direction and has an intensity approximatelyproportional to a known function of the difference between the first andsecond electrical potentials; and the beam combiner comprising a fixedsurface grating;

diode laser means and collimating means for directing input light ofintensity approximately proportional to the coefficient a_(n) to theinput end of the nth optical module;

means for applying across the electrodes of the modulator, while thelight is passing through the modulator, a potential differenceapproximately proportional to a function of x such that the intensity ofthe output light from the modulator is approximately proportional toa_(n) x;

diode laser means and collimating means for directing input light ofintensity approximately proportional to the coefficient a_(n-1), via thebeam combiner, to the output end of the optical module, to combine withthe output light from the modulator so that the intensity of the outputlight from the nth optical module is approximately proportional to a_(n)x+a_(n-1) ;

the (n-1)th optical module being positioned to receive the output lightfrom the nth optical module in the predetermined input direction of the(n-1)th module;

and so on, in the same manner, and finally to the first optical module,so that the intensity of the polynomial function output light from thefirst optical module is approximately proportional to

    ((((a.sub.n x+a.sub.n-1)x+a.sub.n-2)x+ -- -- -- +a.sub.2)x+a.sub.1)x+a.sub.o.

DRAWINGS

All of the drawings are schematic views.

FIG. 1 illustrates the functions of a typical basic calculational moduleused in practicing the present invention. The outputs are the indicatedfunctions of the inputs.

FIG. 2 shows a typical polynomial pipeline processor comprising fourmodules as in FIG. 1 to evaluate a fourth-order polynomial in onevariable, x. Note that x is passed through unchanged.

FIG. 3 shows a typical bulk optical embodiment of a calculational moduleas in FIG. 1.

FIG. 4 shows a typical combination as in FIG. 2 of optical modules as inFIG. 3 and associated components in a polynomial pipeline processoraccording to the invention. The time delays are introduced to compensatefor the propagation delay through the modules.

FIG. 5 illustrates the functions of a typical optical computationalmodules as in FIGS. 3 and 4 using a beam splitter (combiner) ofefficiency η.

FIG. 6 shows a portion of a typical embodiment of an integrated opticalpipeline processor according to the invention utilizing electroopticgrating modulators and surface grating adders (combiners).

FIGS. 7a and b show a typical combination of pipeline processors as inFIGS. 4 and 6. FIG. 7a illustrates the use of the problem-divisiontechnique to accommodate complex, positive a_(k) and complex x of eithersign. FIG. 7b illustrates a further division to accommodate either signfor a_(k).

FIG. 8 shows a typical modification of a processor as in FIG. 6, whereina single laser, beam splitters, and surface acoustic wave modulatorsfurnish all of the light inputs.

FIG. 9 shows another typical modification of a processor as in FIG. 4 orFIG. 6, wherein the same electrical potential is connected to all of themodulators with no time delays.

FIG. 10 illustrates typical circuitry employing a strip line to providethe time delays indicated in FIGS. 4 and 6.

FIG. 11 shows a typical combination of pipeline processors as in FIGS. 4and 6 to accommodate all real values of x, positive, zero, and negative.

FIG. 12 shows a typical array of modules as in FIGS. 1-6 for evaluatingpolynomials in two variables.

FIG. 13 shows another typical array as in FIG. 12 for evaluatingpolynomials in two variables.

CARRYING OUT THE INVENTION

The following disclosure includes a slight revision of the paper by C.M. Verber, R. P. Kenan, H. J. Caulfield, Jacques E. Ludman, and P.Denzil Stilwell, Jr., "Suggested Integrated Optical Implementation ofPipelined Polynomial Processors"; in Advances in Optical InformationProcessing, Proc. SPIE, vol. 388, 1983, pp. 221-227 (Paper Number 388-30of the SPIE (Society of Photo-Optical Instrumentation Engineers) LosAngeles Technical Symposium, held Jan. 17-21, 1983; presented thereorally on Jan. 21, 1983.). In the paper, it is shown that opticalsystolic pipeline processors for polynomial evaluation can be builtusing Horner's rule; and that, with integrated optics techniques, it ispossible to fabricate large-order pipelines operating at very highspeeds.

Polynomial Evaluation Optical Pipelines

The use of optics to evaluate numerical quantities is of great interestbecause the evaluations can often be carried out at high speed withlittle electrical power consumption. Because no general purpose opticalcomputers are available, we must design new algorithms and architecturesfor each new task. Recently optics has begun to adapt methods fromelectronic systolic array processing.sup.(1) for thesepurposes.sup.(2,3). Here we consider optical systolic polynomialevaluation using Horner's rule.sup.(4). We will show that simple opticalpipelines can evaluate polynomials with positive coefficients and thatspatio-temporal arrays of these pipelines can perform far more generaltasks.

For the moment, let us limit ourselves to

polynomials of one variable,

positive variables,

positive exponents, and

positive coefficients.

All of these restrictions can be removed, but these details do not addmuch to this illustration. Consider the polynomial ##EQU2##

The factorization or synthetic division displayed in Eq. (1) is an oldtechnique, known by Newton but usually ascribed to Horner, whichprovides a recursive means for evaluating polynomials. As is shownbelow, it is straightforward to cast the more general case (allowingnegative x's, a's, and exponents) as a sum of calculations that may becarried out in series or in parallel.

Kung.sup.(5) has shown how to pipeline the polynomial evaluation of Eq.(1) using four digital modules. Each module performs the functionsindicated in FIG. 1. It accepts two inputs α and β, and generates twooutputs δ and ε, where

    δ=α

and

    ε=αβ+γ

and where γ is a stored or locally supplied value. The pipelining offour such modules to calculate a 4th order polynomial is shown in FIG.2.

An optical module for carrying out one step in the calculation is shownin FIG. 3. It has a modulator (corresponding to α in FIG. 1), a lens tokeep the light together, a beam splitter functioning as a combiner(addition element), a new modulated light source (corresponding to γ inFIG. 1), and spacers. An input beam of intensity β leads to an outputbeam of the form c[αβ+γ]. Each module is calibrated to produce aconstant c(<1). Thus, we produce, as output from the fourseries-connected optical modules, a signal proportional to C⁴ γ. Forlarge N's, we want c to be as large as possible to keep c⁴ γ intenseenough to detect rapidly.

FIG. 4 shows a typical way to connect these modules. Delay lines oflength τ/4, τ/2, 3τ/4, and τ give an output pair x,p(x) (after a delaytime τ) as fast as we can input x. Modulators capable of 10⁹ Hz arequite practical and much higher modulation frequencies are possible inprinciple. The ability of the system to handle high speed inputs isprobably limited by the accuracy with which the time delays are set.Setting them to 10⁻¹¹ sec (˜one millimeter) is routine.

Optical Implementation of Pipeline Processing As is evident from thepreceding discussion, the basic pipeline processor unit of FIG. 1 mustbe capable of performing the following functions:

Accept inputs x and a'_(n+1) from unit n+1

Store a_(n) or accept it from a local source

Compute a'_(n) '=a'_(n+1) x+a_(n)

Transmit a'_(n) ' and x to unit n-1.

Since there is currently no convenient method for multiplying twooptical intensities, we will take x to be represented by a voltage andthe a_(i) and a'_(i) by optical intensities.

An important consideration in assembling N of the basic units for theevaluation of Nth-order polynomials is to minimize the overall opticalloss of the system. For example, in the arrangement of FIG. 3, we canmaximize the throughput by decreasing the beam-combiner efficienciesη_(i) as we move from left to right through the system. On the otherhand, this would result in greatly increased fabrication difficulty, andthe option of choosing a single η so that identical modules can be usedshould be explored.

If the modules are identical, then the optical intensity correspondingto a_(N-1) will suffer the greatest beam-combiner loss in traversing thesystem [η<0.5 is assumed]. Its intensity after traversing the entireprocessor will be

    I.sub.out =η(η-1).sup.N-1 a.sub.N-1,               (2)

from which it can easily be shown that to maximize I_(out) we set

    η=1/N                                                  (3)

If, for example, N=10, η=0.1 and (1-η)=0.9 corresponding to a loss of0.46 dB per module. This is not an intolerable loss and it is relativelyeasy to make beam combiners with the required efficiency in both bulkand integrated optical configurations.

The most straightforward implementation of the basic computational unitis shown in FIG. 5. The beam combiner is assumed to have an efficiency ηand the modulator to respond to an electrical control signal V=f(x). Toavoid interference effects, the sources from which the opticalintensities representing a_(n+1) and a_(n) are derived must be mutuallyincoherent.

Several factors favor the integrated optical (IO) approach to the devicedesign and fabrication.

(a) We want a high-speed device, so low control voltages V aredesirable. Assuming electrooptic control elements, the IO technology hasan advantage of approximately 10².

(b) In an IO system, reflection problems are minimized. Typically wehave one lens, one modulator and one beam combiner per computationalunit. An entire bulk system must be immersed in an index-matching liquidto keep the losses and stray light due to reflection from the interfacesfrom becoming intolerable. In the IO format, there are essentially nosuch reflection problems. However, there are waveguide losses to contendwith. If we assume a LiNbO₃ waveguide with 0.5 dB/cm loss, then the lossin a 0.5 cm-long computation unit will be 0.25 dB plus, perhaps, anadditional 0.25 dB due to scatter from the components. In the bulk caseif we have six surfaces having 6% loss per surface, the comparablefigure is 1.6 dB per unit.

(c) The production of a device for the evaluation of a high-orderpolynomial requires the fabrication and alignment of a significantnumber of units. Several steps of photolithography are much simpler thanmechanical alignment of individual components. However, in the IO case,coupling of many sources to a single waveguide can be difficult.

A presently preferred configuration for an integrated optical polynomialprocessor is shown in FIG. 6. It is a practical implimentation of theapparatus shown in FIG. 4. The substrate is y-cut LiNbO₃ with a planarsingle-mode Ti-indiffused waveguide. The basic processor unit consistsof an interdigital electrode Bragg modulator.sup.(6) which is energizedby a voltage proportional to x(t), and a fixed surface-grating beamcombiner..sup.(7) Each a_(i) is generated by a diode laser whose outputis collimated by an As₂ S₃ Luneburg lens..sup.(8)

The beam width of the light processing through the processor can bebounded in part by the requirement that some minimal collimation bymaintained without the use of intermediate lenses. This is necessary toensure proper operation of the Bragg gratings. A reasonable startingpoint is to require that the diffraction spread be no greater than 0.1mrad, a condition that is satisfied by a 4 mm beamwidth. This widthplaces no strain on the modulator and adder fabrication, but may beunnecessary severe in terms of grating requirements and lens fabricationtolerances.

The electrooptic modulator comprises an interdigital electrode structurethat is fabricated by standard photolithographic techniques. It is,therefore, desirable to have a line and gap width of at least 1.5 μm,which implies a grating period Λ of at least 6 μm. From the Braggequation,

    sin θ.sub.B =λ.sub.O /2NΛ              (4)

where Λ_(O) is the vacuum optical wavelength and N the mode index of theguided wave, we see that for GaAs-laser wavelengths and N=2.2, θ_(B)≦1.8°. To maintain high modulator performance, even smaller angles, andtherefore larger values of Λ, may be desirable.

It is quite straightforward to fabricate electrooptic grating modulatorsthat achieve their maximum diffraction efficiencies with operatingvoltages of 5 volts or less. However, for reasons that are not wellunderstood, the typical modulator has a maximum diffraction efficiencyof slightly less than 100%. Unless 100% efficiency can be achieved theoperating range 0≦x≦1 will be compromised.

The beam combiners or addition elements are fabricated by theholographic exposure of a suitable photosensitive material to form thegrating, and by a subsequent exposure through a mask to define thelocation and size of the grating region. The presence of the periodicoverlay produces a periodic modulation of the mode index which has theeffect of a thick phase grating on the guided wave. The magnitude of themode-index perturbation is a function of the index of refraction of theoverlay material. If modest diffractive efficiencies are desired, astandard photoresist.sup.(7) can be used. For higher efficiencies As₂S₃.sup.(9) can be used.

The collimating lenses are most conveniently formed as As₂ S₃ Luneburglenses..sup.(8) The Luneburg lens is preferable to the geodesic lensbecause the entire lens array can be formed during a single depositionoperation as opposed to the machining operation required to formgeodesic lenses.

The f/number and focal length of the lens are determined by the desiredwidths of the collimated beam and the divergence of the laser output.For a typical high-quality GaAs single-mode laser the divergenceperpendicular to the plane of the junction is about 35° (full width athalf maximum power). In the waveguide this is reduced by the higherindex to 16°. Therefore, to achieve the previously suggested beam widthof 4 mm, a 14 mm focal length lens is required. The resultant f/numberof f/3.5 is within the current state of the As₂ S₃ Luneburg lensfabrication art.

Orientation of the laser so that its plane is perpendicular to thewaveguide minimizes coupling difficulties and minimizes the lengthneeded for beam expansion. It results in a TM guided wave which requiresbuffer layers under the modulator electrodes to eliminate scattering andabsorptive losses due to the high surface field presented to the metalelectrodes when this polarization is used.

Typical light sources in the design of FIG. 6 are GaAs single-mode diodelasers. It would simplify assembly of the processor if all of the a_(i)could be derived from a single source, with the light distributedthrough a second set of modulators to impart the a_(i) information tothe appropriate beams. The problem with this approach is that, unlessthe beams incident on a given combiner are mutually incoherent,interference effects cause the output of the combiner to be a functionof the phases and amplitudes of the two beams rather than a simple sumof the intensities. In addition, similar interference effects would takeplace at the detector so that the processor output would be highlyambiguous.

The range of wavelengths that can be spanned by the N lasers requiredfor the processor is determined by the properties of the modulators. Inparticular, the diffraction efficiency falls off as the wavelengthdeparts from the design value. The rate at which this occurs is afunction of the grating geometry.sup.(10) and will not be discussed indetail here. The amount of dispersion that can be tolerated isdetermined by the accuracy to which the processor is being designed. Fora 1% accuracy, the acceptable range of wavelength is about 100 A. Thisrequires some laser preselection, but is not too severe a requirement.The tradeoff study required to interrelate these parameters is obviouslyquite complex.

As shown in FIG. 8, an alternative design that allows the use of asingle laser is based on the use of a series of surface acoustic wavesat frequencies f₁, f₂ . . . f_(N), used to impart frequency shifts F₁,f₂ . . . to successive diffracted beams. This produces a set of inputbeams for the processor that are incoherent, but only to a limitedextent, because there will be beats in the output at all possible |f_(i)-f_(j) |. If |f_(i) -f_(j) |=100 MHz, it will be necessary to integratethe output for about 100 nsec to average out the effects of the beats.This will place a severe limitation on the processing speed.

The data input, that is the introduction of x(t), may be accomplished inseveral ways. If the variation of x is slow compared to the timerequired for light to traverse the processor (˜1 nsec) then data can beentered in parallel to all of the modulators, as in FIG. 9, with no lossof speed or accuracy. If, on the other hand, the rate of variation in xis comparable to or faster than the time required for the light totraverse the processor, then a strip line should be used, as in FIG. 10,to match the velocity of the electrical signal to the optical signal byproducing the delays, τ, indicated in FIGS. 4 and 6.

Combinations of Polynomial Pipelines Having described how a basicpolynomial evaluation pipeline can be made, we turn now to the questionof how to combine them to handle less restrictive cases.

First, let us remove the limitation on polynomial order. Suppose thelongest pipeline that can be conveniently fabricated is N=10 and we wantto handle polynomials up to N=20. The Horner alogrithm can be brokenafter any stage with the output from one stage being detected and usedas the input to the next stage. Thus, in this case, we would detect thesignal after stage 10, then re-inject it into a second 10-stagepipeline.

Second, we can remove the restriction to non-negative x's by connectinga pair of pipeline processors as shown in FIG. 11. To allow for all realvalues of x (positive) zero, and negative) we can arrange the polynomialinto odd and even parts. Thus

    P(x)=P.sub.e (x)+P.sub.o (x),                              (5)

where

    P.sub.e (x)=a.sub.0 +a.sub.2 x.sup.2 + . . .               (6)

and

    P.sub.o (x)=a.sub.1 x+a.sub.3 x.sup.3 + . . .              (7)

Let us write

    x=(sgn x)|x|.                            (8)

Then

    P(x)=P.sub.e (|x|)+(sgn x)P.sub.o (|x|).                                  (9)

Thus the real x problem is reduced to two non-negative x problems.

Third, we can remove the restriction to non-negative coefficients. Letus write, for instance,

    P.sub.e (|x|)=P.sub.e.sup.(+) (|x|)-P.sub.e.sup.(-) (|x|) (10)

where

    P.sub.e.sup.(+) (|x|) and P.sub.e.sup.(-) (|x|)                                   (11)

have only non-negative coefficients. To handle all real x's and all reala_(k) 's, we need four pipelines.

Fourth, we can go to complex x's and a_(k) 's. This is straightforwardusing the same technique of dividing the problem we cannot solve intomany problems we can solve (non-negative x's and a_(k) 's). Let anycomplex coefficient be written

    a=R(a)+iI(a),                                              (12)

Then we can write

    P(x)=R(x)+iI(x),                                           (13)

where R(x) and I(x) have only real a's. But each a can still be positiveor negative. The net system looks like FIGS. 7a and b.

Fifth, we can handle polynomials in multiple variables such as

    P(x.sub.1, x.sub.2, . . . ,x.sub.M)=Σa.sub.i,j, . . . m x.sup.i, x.sub.2.sup.j . . . x.sub.M.sup.m                         (14)

by the same method (reduction to easy problems). For example ##EQU3##Given a value for y, a pipeline can be devised for calculating eachb_(m). Clearly this is easily generalized.

For a polynomial of two variables we can write: ##EQU4## It isstraightforward to rewrite this as: ##EQU5## where now the restrictionsare included by allowing b_(mn) to be zero: 15 b_(mn) are non-zero and10 are zero. Then, we must construct the y-polynomials as before, using(generally) 4 pipelines of the form of FIG. 2. This is illustrated inFIG. 12 for the general case and in FIG. 13 specialized to m+n≦4.

Optical function evaluation based upon the concepts described above isso fast that it permits us to search for roots, extrema, derivatives,concidences, etc. systematically, faster than they could be calculateddigitally. Such optical calculators are ideal for real time adaptiveprocessing because they are extremely fast and easily reprogrammed.Integration is simply summation as x is varied (ramped) at a highuniform speed. A non-uniform speed of x variation integrates the productof the polynomial with the modulating function. Steepest descent curvefitting (modulating the different polynomial coefficients at differentfrequencies to obtain the gradient and directly integrating the squareddifference) is easily programmed. Cascades of small-N units can handlemuch larger polynomials.

The optical processor can be implemented in bulk (three-dimensional)modules or in integrated-optics (two-dimensional) modules. The latterapproach involves more initial expense but offers considerableadvantages in cost and compactness of mass-produced optical functioncalculators. This technology also offers ruggedness and permanentalignment with lower power and voltage requirements.

Conclusions

Optical systolic polynomial evaluation appears to be feasible andpromising. Although detailed analyses of speed, accuracy, dynamic range,and related questions have yet to be reported, it is clear that anintegrated optical pipeline can be fabricated and can have a variety ofapplications.

By connecting pipelined polynomial evaluators in series and/or inparallel, we can perform very general operations. The applicationsappear to be quite wide ranging. They include

solving P(x)=0 by ramping x while monitoring P(x)

solving P₁ (x) -P₂ (x)=0,

solving P_(i) (x)=P₂ (x)=P₃ (x)

by seeking coincident zeros of two difference equations,

differentiating at x_(o) by setting x_(o) +ε cos ωt and

monitoring the cos ωt part of P[x(t)], and

integrating by ramping x and time integrating P[x(t)].

(1) H. T. Kung, "Why Systolic Architectures?", Computer, 15, 37-46(1982).

(2) H. J. Caulfield, W. T. Rhodes, M. J. Foster, and Sam Horvitz,"Optical Implementation of Systolic Array Processing", Opt. Com., 40,86-90 (1981).

(3) Jacques E Ludman, H. J. Caulfield, and P. Denzil Stilwell, Jr.,"Robust Optical Systolic Long-Code Processor", Opt. Eng., 21 833-836(1982).

(4) A. Ralston, A First Course in Numerical Analysis (McGraw-Hill, NY,1965) pp 232-235.

(5) H. T. Kung, "Notes on VLSI Computation" Technical Report VLSIDocument V080, Carnegie-Mellon University, Dept. of Computer Science,Pittsburgh, PA 15213 (September 1980).

(6) J. M. Hammer and W. Phillips, "Low-Loss Single-Mode OpticalWaveguides and Efficient High-Speed Modulators of LiNb_(x) Ta_(1-x) O₃on LiTaO₃ ", Appl. Phys. Lett., 24 545-547 (1974).

(7) N. F. Hartman, C. M. Verber, and C. M. Chapman, "Fabrication of a16-Channel Integrated Optical Data Preprocessor", IEEE Trans. onComponents, Hybrids and Manufacturing Technology, CHMT-4 327-331 (1981).

(8) James R. Busch, Van E. Wood, Duncan T. Moore, and W. H. Southwell,"Rectangular Luneburg-Type Lenses for Integrated Optics", Opt. Let.

(9) T. Suhara, Y. Handa, H. Nishihara and J. Koyama, "MonolithicIntegrated Microgratings and Photodiodes for Wavelength Demultiplexing",Appl. Phys. Lett., 40 120-122 (1982).

(10) Herwig Kogelnik, "Coupled Wave Theory for Thick Hologram Gratings",Bell Syst. Tech. J. 48, 2909-2947 (1969).

Referring now to FIG. 4, a typical method according to the presentinvention for providing an optical analog quantity that is approximatelyproportional to the value of a polynomial function expressible in theform ##STR2## comprises directing input light a₄, as via a lens 21, ofintensity approximately proportional to the coefficient a_(n) to theinput end of an nth optical module comprising at the input end amodulator 22 whose output light intensity is approximately proportionalto a known function of an electrical potential difference across it andat the output end a beam combiner 23, applying across the modulator 22,while the light is passing through it, a potential difference, asindicated at 24' (versus ground), approximately proportional to afunction of x such that the intensity of the output light from themodulator 22 is approximately proportional to a a_(n) x, directing theoutput light from the modulator 22, through the beam combiner 23, to theoutput end of the optical module, directing input light a₃, as via alens 25, of intensity approximately proportional to the coefficienta_(n-1), via the beam combiner 23, to the output end of the opticalmodule, to combine with the output light from the modulator 22 so thatthe intensity of the output light from the nth optical module isapproximately proportional to a_(n) x+a_(n-1) ; directing the outputlight from the nth optical module to the input end of an (n-1)th opticalmodule essentially similar to the nth, applying across the modulator 22'in the (n-1)th optical module, while the light received from the nthmodule is passing through it, a potential difference, as indicated at24' (versus ground), approximately proportional to a function of x suchthat the intensity of the output light from the modulator 22' isapproximately proportional to (a_(n) x+a_(n-1))x, directing the outputlight from the modulator 22', through the beam combiner 23', to theoutput end of the optical module, directing input light a₂, as via alens 25', of intensity approximately proportional to the coefficienta_(n-2), via the beam combiner 23, to the output end of the opticalmodule, to combine with the output light from the modulator 22' so thatthe intensity of the output light from the (n-1)th optical module isapproximately proportional to (a_(n) x+a_(n-1))x+a_(n-2) ; and so on, inthe same manner, and finally to the first optical module so that theintensity of the polynomial function output light p(x) from the firstoptical module is approximately proportional to

    ((((a.sub.n x+a.sub.n-1)x+a.sub.n-2)x+ -- -- -- +a.sub.2)x+a.sub.1)x+a.sub.o.

Typically the light of intensity approximately proportional to a_(n),a_(n-1), a_(n-2), ------, a₁, a₀ is substantially monochromatic andmutually incoherent. Typically the modules comprise either bulk opticalcomponents or integrated optical components, and the modulators areeither electrooptic or acoustooptic. Preferably each modulator 22,22',etc comprises an electrooptic grating in a planar waveguide 26, as inFIG. 6.

As shown in FIGS. 4,6, and 10, the potential difference at 24,24', etcapplied across each modulator 22,22', etc is connected via time delaycircuitry 27 such that each potential difference is applied to eachindividual modulator while the light received from the adjacent module,and that had passed through its modulator while the same potentialdifference had been applied across it, is passing through the modulator.Where conditions permit, however, the same potential difference 24 maybe applied across all of the modulators 22,22', etc simultaneously, asin FIG. 9, and remains within small enough limits during the timerequired for light to traverse all of the optical modules as to provideintensities to the polynomial function output light p(x) from the firstmodule within a predetermined desired accuracy.

Where identical modules are to be used throughout a pipeline, each beamcombiner 23,23', etc preferably has an efficiency of about 1/n. Whereoverall optical loss is to be minimized, then preferably the beamcombiner in the nth module has an efficiency of 1/2; the beam combinerin the (n-1)th module has an efficiency of 1/3; the next, 1/4; and soon, 1/5, 1/6, etc; to the beam combiner in the first module having anefficiency of 1/n+1.

Typically the modules are arranged in series to form an integral block;and the modules typically are either bulk components, each contiguous tothe next, as in FIG. 4, or integrated components in a single waveguide26, as in FIG. 6.

To deal with negative values, complex functions, a plurality ofvariables, and other functions requiring more than one block, aplurality of blocks are responsively arranged either in spatial ortemporal combination, as in FIGS. 7, 11, 12, and 13. Typically thepolynomial function output light from at least one block is directed todetermine the intensity of at least one light input to another block, asin FIGS. 7, 12, and 13; and the polynomial function output light of atleast one block typically determines the intensity of at least onecoefficient input light of a block to which it is directed.

As illustrated in FIG. 6, typical apparatus according to the presentinvention for providing an optical analog quantity that is approximatelyproportional to the value of a polynomial function expressible in theform ##STR3## comprises an electrooptic planar waveguide 26; aplurality, n+1, of integrated optical modules in the waveguide 26, eachmodule comprising at its input end a modulator 22,22', etc for receivinglight travelling in a predetermined input direction 31 and transmittinga controlled proportion thereof further in a predetermined outputdirection 32 to the output end, and a beam combiner 23,23', etc forreceiving the light from the modulator 22, etc and transmitting apredetermined proportion thereof further in the predetermined outputdirection 32 and on through the output end of the module; the modulator22, etc comprising electrooptic diffractive means comprising a pair ofelectrodes 33,34 for forming a Bragg grating therebetween, positionedwith a direction of Bragg incidence approximately in the predeterminedinput direction 31, the first electrode 33 comprising a first set ofsubstantially straight and parallel, thin, elongate, electricallyconductive members connected together at one end, and the secondelectrode 34 comprising a second set of substantially straight andparallel, thin, elongate, electrically conductive members, interleavedwith the first set, insulated therefrom, and connected together at theopposite end, so that when a first electrical potential (ground) isapplied to the first electrode 33 and a second electrical potential24,24' etc is applied to the second electrode 34 the controlled portionof input light transmitted through the modulator 22, etc is provided bya Bragg diffraction thereof in the predetermined output direction 32 andhas an intensity approximately proportional to a known function of thedifference between the first and second electrical potentials; and thebeam combiner 23 comprising a fixed surface grating; diode laser meansa₄ and collimating means 21 for directing input light of intensityapproximately proportional to the coefficient a_(n) to the input end ofthe nth optical module; means for applying, as at 24, across theelectrodes 33,34 of the modulator 22, while the light is passing throughthe modulator 22, a potential difference approximately proportional to afunction of x such that the intensity of the output light from themodulator is approximately proportional to a_(n) x; diode laser means a₃and collimating means 25 for directing input light of intensityapproximately proportional to the coefficient a_(n-1), via the beamcombiner 23, to the output end of the optical module, to combine withthe output light from the modulator 22 so that the intensity of theoutput light from the nth optical module is approximately proportionalto a_(n) x+a_(n-1) ; the (n-1)th optical module being positioned toreceive the output light from the nth optical module in thepredetermined input direction 31 of the (n-1)th module; means forapplying, as at 24', across the electrodes 33,34 of the modulator 22' inthe (n-1)th optical module, while the light received from the nth moduleis passing through the modulator 22', a potential differenceapproximately proportional to a function of x such that the intensity ofthe output light from the modulator is approximately proportional to(a_(n) x+a_(n-1))x; diode laser means a₂ and collimating means 25' fordirecting input light of intensity approximately proportional to thecoefficient a_(n-2), via the beam combiner 23', to the output end of theoptical module, to combine with the output light from the modulator 22'so that the intensity of the output light from the (n-1)th opticalmodule is approximately proportional to (a_(n) x+a_(n-1))x+a_(n-2) ; andso on, in the same manner, and finally to the zeroth optical module, sothat the intensity of the polynomial function output light p(x) from thezeroth optical module is approximately proportional to

    ((((a.sub.n x+a.sub.n-1)x+a.sub.n-2)x+ -- -- -- +a.sub.2)x+a.sub.1)x+a.sub.o.

Typically the waveguide 26 comprises a planar single mode waveguidecomprising titanium diffused into a surface of a substrate comprisingy-cut lithium niobate, each diode laser means a₄,a₃, etc comprises agallium arsenide single-mode laser, and each collimating means 21,25,etc comprises a Luneburg lens, which typically comprises arsenictrisulfide. Preferably the f/number of the lens is about f/3 to f/6, andthe focal length is about 15 to 30 millimeters.

Typically each diode laser means a₄, etc is positioned with the plane ofthe diode junction approximately perpendicular to the plane of thewaveguide 26, and each modulator 22, etc includes, between the waveguide26 and the electrodes 33,34, a transparent electrically insulatingbuffer layer of thickness less than about one-tenth of the spacingbetween adjacent conductive members of each electrode 33,34.

Preferably the light of intensity approximately proportional to a_(n),a_(n-1), a_(n-2), -- -- -- , a₁, a₀ is substantially monochromatic andmutually incoherent. Typically each diode laser means a₄, etc provideslight of a different wavelength, all of the wavelengths preferably beingwithin a range of about 100 Angstroms. In some embodiments of theinvention, as shown in FIG. 8, all of the diode laser means are formedby a single laser 21 and surface acoustic wave means f₄,f₃, etc whichshift the frequency of the light directed to each optical module so thatthe frequency is different for each module; and means are included forintegrating the output of each module for at least about ten cycles ofthe minimum difference frequency, to provide an output responsive to theaverage intensity of the components at the various frequencies that arepresent.

As shown in FIGS. 6 and 10, the potential difference at 24,24', etcapplied across the electrodes 33,34 of each modulator 22,22', etc isconnected via time delay circuitry 27 such that each potentialdifference is applied to each individual modulator while the lightreceived from the adjacent module, and that had passed through itsmodulator while the same potential difference had been applied acrossit, is passing through the modulator. Where conditions permit, however,the same potential difference 24 may be applied across the electrodes33,34 of all of the modulators 22, etc simultaneously, as in FIG. 9, andremains within small enough limits during the time required for light totraverse all of the optical modules as to provide intensities of thepolynomial function output light p(x) from the first module within apredetermined desired accuracy.

Where identical modules are to be used throughout a pipeline, each beamcombiner 23,23', etc preferably has an efficiency of about 1/n. Whereoverall optical loss is to be minimized, then preferably the beamcombiner in the nth module has an efficiency of 1/2; the beam combinerin the (n-1)th module has an efficiency of 1/3; the next, 1/4; and soon, 1/5, 1/6, etc; to the beam combiner in the first module having anefficiency of 1/n+1.

Typically the modules are arranged in series to form an integral block;and the modules typically are integrated components in a singlewaveguide 26, as in FIG. 6.

To deal with negative values, complex functions, a plurality ofvariables, and other functions requiring more than one block, aplurality of blocks are responsively arranged either in spatial ortemporal combination, as in FIGS. 7a, 7b, 11,12, and 13. Typically thepolynomial function output light from at least one block is directed todetermine the intensity of at least one light input to another block, asin FIGS. 7a, 7b, 12, and 13; and the polynomial function output light ofat least one block typically determines the intensity of at least onecoefficient input light of a block to which it is directed.

I claim:
 1. Apparatus for providing an optical analog intensity that isapproximately proportional to the value of a polynomial functionexpressible in the form ##STR4## wherein p(x) is a function of x,x isany variable, n is any positive integer, the cooefficients, a_(i), areany constants, variables, or constants and variables;comprising: anelectrooptic planar waveguide; a plurality, n+1, of integrated opticalmodules in the waveguide, each module comprising at its input end amodulator for receiving light travelling in a predetermined inputdirection and transmitting a controlled proportion thereof further in apredetermined output direction to the output end, and a beam combinerfor receiving the light from the modulator and transmitting apredetermined proportion thereof further in the predetermined outputdirection and on through the output end of the module; the modulatorcomprising electrooptic diffractive means comprising a pair ofelectrodes for forming a Bragg grating therebetween, positioned with adirection of Bragg incidence approximately in the predetermined inputdirection, the first electrode comprising a first set of substantiallystraight and parallel, thin, elongate, electrically conductive membersconnected together at one end, and the second electrode comprising asecond set of substantially straight and parallel, thin, elongate,electrically conductive members, interleaved with the first set,insulated therefrom, and connected together at the opposite end, so thatwhen a first electrical potential is applied to the first electrode anda second electrical potential is applied to the second electrode thecontrolled portion of input light transmitted through the modulator isprovided by a Bragg diffraction thereof in the predetermined outputdirection and has an intensity approximately proportional to a knownfunction of the difference between the first and second electricalpotentials; and the beam combiner comprising a fixed surface grating;diode laser means and collimating means for directing input light ofintensity approximately proportional to the coefficient a_(n) to theinput end of the nth optical module; means for applying across theelectrodes of the modulator, while the light is passing through themodulator, a potential difference approximately proportional to afunction of x such that the intensity of the output light from themodulator is approximately proportional to a_(n) x; diode laser meansand collimating means for directing input light of intensityapproximately proportional to the coefficient a_(n-1), via the beamcombiner, to the output end of the optical module, to combine with theoutput light from the modulator so that the intensity of the outputlight from the nth optical module is approximately proportional to a_(n)x+a_(n-1) ; the (n-1)th optical module being positioned to receive theoutput light from the nth optical module in the predetermined inputdirection of the (n-1)th module; means for applying across theelectrodes of the modulator in the (n-1)th optical module, while thelight received from the nth module is passing through the modulator, apotential difference approximately proportional to a function of x suchthat the intensity of the output light from the modulator isapproximately proportional to (a_(n) x+a_(n-1))x; diode laser means andcollimating means for directing input light of intensity approximatelyproportional to the coefficient a_(n-2), via the beam combiner, to theoutput end of the optical module, to combine with the output light fromthe modulator so that the intensity of the output light from the (n-1)thoptical module is approximately proportional to (a_(n)x+a_(n-1))x+a_(n-2) ; and so on, in the same manner, and finally to thezeroth optical module, so that the intensity of the polynomial functionoutput light from the zeroth optical module is approximatelyproportional to

    ((((a.sub.n x+a.sub.n-1)x+a.sub.n-2)x+ -- -- -- +a.sub.2)x+a.sub.1)x+a.sub.o.


2. Apparatus as in claim 1, wherein the waveguide comprises a planarsingle mode waveguide comprising titanium diffused into a surface of asubstrate comprising y-cut lithium niobate.
 3. Apparatus as in claim 1,wherein each diode laser means comprises a gallium arsenide single-modelaser.
 4. Apparatus as in claim 3, wherein each collimating meanscomprises a Luneburg lens.
 5. Apparatus as in claim 4, wherein the lenscomprises arsenic trisulfide.
 6. Apparatus as in claim 4, wherein thef/number of the lens is about f/3 to f/6, and the focal length is about15 to 30 millimeters.
 7. Apparatus as in claim 1, wherein each diodelaser means is positioned with the plane of the diode junctionapproximately perpendicular to the plane of the waveguide.
 8. Apparatusas in claim 7, wherein each modulator includes, between the waveguideand the electrodes, a transparent electrically insulating buffer layerof thickness less than about one-tenth of the spacing between adjacentconductive members of each electrode.
 9. Apparatus as in claim 1,wherein the light of intensity approximately proportional to a_(n),a_(n-1), a_(n-2), -- -- -- , a₁, a₀ is substantially monochromatic andmutually incoherent.
 10. Apparatus as in claim 9, wherein each diodelaser means provides light of a different wavelength, all of thewavelengths being within a range of about 100 Angstroms.
 11. Apparatusas in claim 9, wherein all of the diode laser means are formed by asingle laser and surface acoustic wave means which shift the frequencyof the light directed to each optical module so that the frequency isdifferent for each module.
 12. Apparatus as in claim 11, comprisingmeans for integrating the output of each module for at least about tencycles of the minimum difference frequency, to provide an outputresponsive to the average intensity of the components at the variousfrequencies that are present.
 13. Apparatus as in claim 1, wherein thepotential difference applied across the electrodes of each modulator isconnected via time delay circuitry such that each potential differenceis applied to each individual modulator while the light received fromthe adjacent module, and that had passed through its modulator while thesame potential difference had been applied across it, is passing throughthe modulator.
 14. Apparatus as in claim 1, wherein the same potentialdifference is applied across the electrodes of all of the modulatorssimultaneously and remains within small enough limits during the timerequired for light to traverse all of the optical modules as to provideintensities of the polynomial function output light from the firstmodule within a predetermined desired accuracy.
 15. Apparatus as inclaim 1, wherein each beam combiner has an efficiency of about 1/n. 16.Apparatus as in claim 1, wherein the beam combiner in the nth module hasan efficiency of 1/2; the beam combiner in the (n-1)th module has anefficiency of 1/3; the next, 1/4; and so on, 1/5, 1/6, etc.; to the beamcombiner in the first module having an efficiency of 1/n+1. 17.Apparatus as in claim 1, wherein the modules are arranged in series toform an integral block.
 18. Apparatus as in claim 17, wherein aplurality of blocks are responsively arranged either in spatial ortemporal combination.
 19. Apparatus as in claim 18, wherein thepolynomial function output light from at least one block is directed todetermine the intensity of at least one light input to another block.20. Apparatus as in claim 19, wherein the polynomial function outputlight of at least one block determines the intensity of at least onecoefficient input light of a block to which it is directed.